three-dimensional mapping and range measurement by means of projected speckle patterns

Three-dimensional mapping and range measurementby means of projected speckle patterns

Javier García,1,* Zeev Zalevsky,2 Pascuala García-Martínez,1 Carlos Ferreira,1

Mina Teicher,3 and Yevgeny Beiderman3

1Departamento de Óptica, Universitat de València, C/Dr. Moliner, 50, 46100 Burjassot, Spain2School of Engineering, Bar-Ilan University, Ramat-Gan 52900, Israel

3Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

*Corresponding author: [email protected]

Received 15 November 2007; revised 27 March 2008; accepted 9 May 2008;posted 9 May 2008 (Doc. ID 89725); published 26 May 2008

We present a novel approach for three-dimensional (3D) measurements that includes the projection ofcoherent light through ground glass. Such a projection generates random speckle patterns on the objector on the camera, depending if the configuration is transmissive or reflective. In both cases the spatiallyrandom patterns are seen by the sensor. Different spatially random patterns are generated at differentplanes. The patterns are highly random and not correlated. This low correlation between different pat-terns is used for both 3D mapping of objects and range finding. © 2008 Optical Society of America

OCIS codes: 030.6140, 100.6890, 110.6150.

1. Introduction

Three-dimensional (3D) mapping and ranging are re-quired in computer vision, automation, and opticalimage processing. Multiple techniques have been de-vised to accomplish this task [1]. A common principleof many methods is triangulation, using either con-ventional illumination [2–4] or coherent illumination[5,6]. Themain problemwith triangulation is the pre-senceof shadowsthat inhibit themappingofsteepsur-faces. Other multiple optical 3D mapping methodsinclude coherence radar [7] and speckle pattern sam-pling [8]. In both cases the experimental setup is rela-tively complex due to the interferometric nature of thetechnique in the first case and the need of a tunablelaser in the second case. Holography can also be usedfor 3D mapping (or, rather, contouring) by the use ofmultiple wavelengths [9], multiple sources [9], orobject displacement between two records [10].A significant part of the research is based or condi-

tioned by the speckle patterns that happen when

coherent light is reflected or transmitted froma roughsurface [11–13]. These patterns are the result of inter-ference among scatteredwavelets, each arising fromadifferent microscopic element of the rough surface.Speckle patterns have the remarkable quality of eachindividual speckle serving as a reference point fromwhich changes in the phase of the light scattered fromthe surface can be tracked. Because of this, speckletechniques such as electronic speckle pattern inter-ferometry (ESPI) have been widely used for displace-ment measuring and vibration analysis [14–17]. Thisprocedure produces correlation fringes that corre-spond to the object’s local surface displacementsbetween two exposures. In speckle interferometrictechniques the changes in the speckle pattern is re-vealed by adding a coherent reference beam, whichconverts the subtle variations of phase into a differentintensitydistribution. It is also possible to analyze thechanges in the intensity of the speckle pattern. Underthe proper conditions a small rigid body motion of thesample that is illuminated with coherent light willmainly displace the diffracted speckle pattern withlittle change in thedistribution of intensity aside fromthe shift. This fact enables the use of speckle photo-

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3032 APPLIED OPTICS / Vol. 47, No. 16 / 1 June 2008

graphy for analyzing displacements and rotations bycomparing the shift between two intensity specklepatterns captured in two different situations[11,14,17,18]. It is worth noting that speckle photo-graphy and interferometry by themselves can mea-sure the distortion of an object but not its actual 3Dshape or range.For the most part, all the above-referred methods

related to speckle patterns used the calculation of thecross-correlation function for the speckle pattern be-fore and after the deformation of an object. All ofthese methods try to avoid the speckle decorrelation,which is the single most important limiting para-meter in measuring systems based on laser speckles,and many efforts have been made to reduce it.This paper is focused on using the decorrelation be-

tweendifferent speckle patterns. Themethod is basedon the study of the speckle patterns projected on a cer-tain object. We propose to use a diffuse object, such asa ground glass, to project a speckle pattern onto theobject. Inherent changes of the speckle pattern as itpropagates will uniquely characterize each locationin the working volume. Notably it is the projectedspeckle pattern that has to be sensed and analyzed,not the speckle pattern produced by the object itself.We apply these speckle pattern changes to three dif-ferent applications of 3D mapping for both transmis-sive objects and diffusively reflecting objects andrange estimation. We determine the map of a 3D ob-ject or its range, observing the differences and simila-rities of the speckled patterns, when an object isilluminated using light coming from a diffuser. Thecorrelation between the initial and final patterns de-creases with the object displacement, so if the displa-cement is significant, the speckle patterns arestatistically uncorrelated. Because of this we first ca-librate the region of displacements as a reference todetermine the profile of a certain 3D object. To ourknowledge, range estimation has never been mea-sured using speckle patterns. In contrast to other pat-tern projection or triangulation techniques, the 3Dresolution in this approachdoesnot dependon thedis-tance or relative angle between the camera and thepattern projector, and thus it is affected by shadowingproblems. Moreover the system is robust and reliableas the only interferometric process is concentratedonly on the speckle pattern generation, and it doesnot depend on the object accurate positioning.

2. Theoretical Background of the Projected SpecklePatterns

As stated in Section 1, we are dealing with the speck-les projected by the light transmitted by a diffuser.Thus we will need for our applications a descriptionof the speckle pattern in free-space propagation, inparticular, the axial and transverse correlation prop-erties. Also, we will use a configuration where a dif-fuse object illuminated with a speckle pattern isitself imaged by an optical system. Although thefundamental equations are displayed, we refer to

theReferences (mainly [11–13]) for a complete deriva-tion and general discussion.

Let us first consider the description of the specklepatterns and their evolutionwhen the object is axiallytranslated. The derivation is based fully on tempo-rally coherent radiation. This condition can be easilyattained if the laser used has a coherence length long-er than the maximum path length difference in thesetup, a situation that occurs in the later analyzed set-ups. A ground glass object, gðx; yÞ (Plane G) normal tothe optical axis (see Fig. 1), is transilluminated by amonochromatic parallel beam of laser light of wave-length λ. Speckle patterns are recorded at Plane C,parallel to the object plane. In Plane C, at a distancez from Plane G, the amplitude distribution obtainedby the Fresnel diffraction equation is proportionalto [19]

Uðξ; ηÞ ¼ZZ

gðx; yÞ

× exp�jπλz

hðξ − xÞ2 þ ðη − yÞ2

i�dxdy; ð1Þ

and the irradiance is given by

Iðξ; ηÞ ¼��ZZ

gðx; yÞ exp�jπλz

hx2 þ y2

i�

× exp�−j

2πλz

hξxþ ηy

i�dxdy

��2: ð2Þ

To check the change due to propagation, assume thatthe diffuser is axially translated a distance Δz (seeFig. 1). The irradiance at Plane C is given byEq. (2), replacing z with z − z0 as [19]

Iðξ; ηÞ ¼��RR gðx; yÞ exp

�j πλz ½x2 þ y2�

�

× exp�jπΔz

λz2 ½x2 þ y2��exp

�−j

2πλz

hξ�1þΔz

z

�x

þ η�1þΔz

z

�yi�

dxdy��2: ð3Þ

Fig. 1. Variation of the speckle pattern in Plane G when theground object is axially shifted through Δz.

1 June 2008 / Vol. 47, No. 16 / APPLIED OPTICS 3033

The interpretation of Eq. (3) shows that the axialtranslation of g has two main effects [19]. One effectis a radial shift of the speckles linked to the magnifi-cation due to geometrical projection from the diffuserscreen center to the recording plane. The other effectis a decorrelation of the corresponding speckles due tothe term expðj πΔz

λz2 ½x2 þ y2�Þ.The first-order statistic of intensity shows that the

intensity obeys an exponential law with a mean (andstandard deviation) equal to the mean intensity inthe case of fully developed speckle. For this probabil-ity law to hold, the distribution of phases in the dif-fuser must be uniform in the ð0; 2πÞ interval, which iseasily achieved in most ground glass diffusers (asshown in Chap. 4 of Ref. [12]) and is enough thatthe standard deviation of phases will be in the orderof the wavelength.In the context of this paper, the second-order sta-

tistics play a fundamental role. They can be derivedfrom the transverse correlation coefficient of the fieldat two positions of the pattern [12,13,20] that, asidefrom constant factors, for the free-space propagationcase and for a sufficiently fine grain diffuser is

ΓATðΔξ;ΔηÞ ¼ZZ þ∞

−∞

Iðx; yÞ

× exp�−j

2πλ�xΔξþ yΔη

��dxdy: ð4Þ

The correlation function of the intensity betweentwo points separated in the transverse plane by adistance ðΔξ;ΔηÞ is

ΓITðΔξ;ΔηÞ

¼ �I2�1þ

��RRþ∞

−∞

Iðx; yÞ exp½−j 2πλ ðxΔξþ yΔηÞ�dxdyRRþ∞

−∞

Iðx; yÞdxdy��;

ð5Þ

where Iðx; yÞ is the intensity in the diffuser (which wetake as constant over its extent) and �I is the mean ofthe intensity in the output plane. Note that with theproper scaling, the correlation function of intensity isgoverned by the Fourier transform of the diffusershape. In our later-described experimental system,we use a circular uniform-intensity diffuser. For thiscase Eq. (5) reduces to

ΓITðrÞ ¼ �I2"1þ 2

��J1

�πΦsλz

�πΦsλz

��#; ð6Þ

whereΦ is the diameter of the diffuser, J1 is the first-kind Bessel function, and s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔξ2 þΔη2

pis the

distance between the two sampled points in thetransverse plane at distance z from the diffuser.Not so surprisingly the correlation follows the samelaw (except for additive andmultiplicative constants)as the diffraction spot obtained by an imaging system

[21]. Eventually the correlation, which is related tothe speckle size, is determined by the numericalaperture of the diffuser as seen at the observationplane. At large distances between the samples, thecorrelation decreases to a background level, corre-sponding to the autocorrelation of the average inten-sity value. The speckle lateral size can be obtained byreasoning that inside a given speckle, the correlationmust be relatively high. We can take as criterion,analogously to the Rayleigh criterion, the distancewhere the Airy pattern takes its first minimum.Then we obtain the average transverse size (radius)of the speckle pattern as

ST ¼ 1:22λzΦ : ð7Þ

As in any resolution criterion, the factor in front ofEq. (7) is somehow arbitrary and depends on the con-vention for speckle size. It is worth recalling that thisis a statistical estimation value, and while it fixes thesmallest size of the pattern features, the correlationcan often exceed this length.

With respect to the longitudinal extent of thespeckles, the problem reduces again to the calcula-tion of the axial correlation function of the intensity[13,20] between two points separated axially by Δz.In this case, with the same assumptions as the trans-verse case plus the requirement thatΔzwill be smallcompared with z, the correlation of intensity resultsin

ΓILðΔzÞ ¼ �I2�1þ 2

��sin c

� Φ2

8λz2 Δz

��: ð8Þ

Again the expression, except for the bias correspond-ing to the correlation of the intensity average, coin-cides with that of the axial resolution of an imagingsystem (see Section 8.8 of Ref. [21]). Taking the firstnull of the sinc function as the criterion to define thespeckle size, the half width of the speckle is

SL ¼ 8λ�zΦ

�2

; ð9Þ

where the 8 factor is also subject to the criterion de-finition. A simplified picture of the speckle pattern inthe volume can be visualized by a set of spots at ran-dom locations with transverse and longitudinal sizegiven by Eqs. (7) and (9), respectively. Note that thetransverse and longitudinal sizes increase linearlyand quadratically with the axial distance, respec-tively. In fact, for large z compared with the trans-verse dimensions of the experiment, the specklesare elongated along lines radiating from the diffusercenter.

The axial and transverse correlations have only amain maximum around the origin, denying the ap-pearance of long distance correlations. This is the


correlation at a long distance compared with thespeckle size due only to the average intensity value.As amain conclusion two speckle patterns taken at

lateral or axial distances larger than the transverseor axial speckle sizes [Eqs. (7) and (9)] will be uncor-related.The second situation we need to analyze is the ima-

ging configuration in which a diffuse object, coher-ently illuminated, is imaged by means of a lens.With the condition that both the correlation area onthe diffuser and the lens resolution spot size are smallcompared with the illuminated area, the same equa-tions are valid anda fully developed speckle pattern isobtained in the image plane.We just consider the lensitself a perfect diffuser for speckle characteristics cal-culations [11,12]. Then the image speckle size is de-termined by the numerical aperture of the imaginglens or,more directly, by its resolution spot size,whichwill coincide with the speckle size in the image.A case where the diffuse object is itself illuminated

by a speckle pattern is a particular case of the above-described imaging configuration (eventually, a fixedspeckle pattern is a coherent illumination). If thespeckle size of the pattern projected onto the objectis much larger than the lateral typical size of the sur-face microstructure, the phase of the projected pat-tern can be neglected (as it is randomized by thediffuser), and the problem is analogue to a diffusiveobject with intensity reflectance given by the pro-jected pattern and random phase given by the objectmicrostructure.Of particular interest for this paper isthe case where the system resolution spot size on theobject is significantly smaller than the projected spotsize. Then the image can be seen as the image of theintensity of the projected speckle pattern (unaffectedas its resolution is much smaller than the system re-solution) modulated by a fine speckle pattern withcharacteristics given by the lens aperture.Wewill de-note this speckle pattern as the imaging speckle todistinguish it from the primary speckle pattern pro-jected onto the object.Finally we consider the situation where the lens

images a plane at a distance from the diffuser (seeFig. 2). Now the lens makes the coherent imagingof the complex speckle pattern produced by free-spacepropagation. The spatial frequencies characteristicsof the speckle pattern at a distance z from the diffuserin the fully developed speckle case is given by thepower spectral density, which is just the Fourier

transform of the intensity autocorrelation [Eq. (5)].Although this is the most common statistical descrip-tion, we are interested in coherent imaging, so the re-levant function is the Fourier transform of thecoherence function of the amplitude [Eq. (4)], whichis itself the Fourier transform of the diffuser shape.The spatial frequencies are limited by free-spacepropagation to a cutoff frequency:

νc ¼Φ2λz : ð10Þ

The coherent transfer function of an imaging systemis also a circ function. Therefore, provided that thecutoff frequency of the optical system is larger thanthat of the speckle pattern, the effect of imaging isonly scaling. This is always accomplished with theangle of the diffuser, subtended from the object, beingsmaller than the angle of lens.

3. Experimental Results for Transmission ThicknessMapping

In Fig. 2 we show the system for the 3D mapping oftransparent objects. We project laser light on a smallcircle on ground glass with an adjustable diameter bymeans of an iris diaphragm. This illumination cre-ates a speckle pattern in the volume after it, withthe illumination distribution depending on the dis-tance from the ground glass. The speckle in a certainplane, at distance z from the ground glass, is imagedon a CCD camera. The diaphragm is adjusted untilthe size of the speckle on the CCD equals a few pix-els, giving a distinct speckle pattern image. The Fnumber of the camera is about 4 (resolution is wellbelow the camera pixel size), so it does not influencethe speckle pattern, except for magnification.

The insertion of a transparent plane parallel platebetween the camera and the diffuser changes theplane that is imaged in the camera by a distanceΔz that depends on the thickness and the index ofthe plate. As explained in Section 2, the speckleslying in the image plane suffer a decorrelation whenthe light path increases more than the averagelength of speckles (SL) in the z direction [Eq. (9)].

We first calibrate the system by mapping thespeckles for the thicknesses of interest. Then, afterplacing an object composed of patches of plane par-allel plates of different thicknesses and imagingthe generated speckles pattern, we estimate its 3Dmapping. The calibration plates (and those used inthe later experiment) are set perpendicular to the op-tical axis to null the lateral displacement of thespeckle pattern. The thickness of the plates can alsointroduce a change in the scale. Although this effectis inherently taken into account in the calibrationprocess, we use a telecentric lens to reduce the scalechanges.

In Fig. 3 we show some of the images used for thecalibration of the system. In Fig. 3(a) we present thecaptured speckle pattern when no transmissive ob-ject is placed. In Fig. 3(b) we placed, instead of theFig. 2. Setup for transmissive object 3D mapping.


object, glass with a thickness of 0:5mm. In Figs. 3(c)and 3(d) we placed glass with a thickness of 1mmand 1:5mm, respectively. As one may see from theobtained images, the patterns are neatly different.The dark vertical lines on the figures are the shadowof the border of the plate, with the left band of theimage remaining uncovered (note that it remainsunaffected along the figures). This fact reduces theuseful area but permits the registering of the subse-quent image. The refraction index of the glass usedas the inspected object was 1.48. The laser used forthe experiment was a He–Ne laser with wavelengthof 633nm. The F number of the diffuser, as seen fromthe plates plane, is approximately 8. This gives anaxial resolution (for full pattern decorrelation) of300 μm. The additional axial path length introducedby a glass plate is ðn − 1Þ times the thickness. There-fore for 0:5mm of glass we get 240 μm of addedoptical path, sufficient for producing an almost fulldecorrelation.InFig. 4we constructed anobject containing spatial

regions with no glass, glass with a 0:5mmwidth, andglasswith a1mmwidth. This is denoted inFig. 4(a) asregions 0, 1, and 2, respectively. Figure 4(b) shows theimage captured in the CCD camera. Figure 5 showsthe reconstruction results while the reconstructionis done by the absolute value of the subtraction be-tween the captured patterns and the reference pat-terns (obtained in the calibration process). Thedark zones indicate detection. In Fig. 5(a) we subtractthe reference pattern of zerowidth glass in the opticalpath. In Figs. 5(b) and 5(c), we show the subtraction ofthe 0:5mmwidth and 1mmwidth glass slides. As onemay see the dark regions appeared where they weresupposed to, following the spatial construction of the

object as indicated in Fig. 4(a). Note that the leftmostpart of Region 1 [see Fig. 5(b)] is not detected becausethe calibration plates did not cover this left band.

This method permits the 3D mapping of staircase-type objects. The main restriction on the object isthat it must be composed of plane parallel plates.Even a small wedge would divert the light prevent-ing its fall on the lens aperture. A tilt on the objectwould, in addition to the decorrelation due to the in-spection of a different axial plane (it introduces long-er optical path), introduce a shift in the pattern thatwould require a spatial registering of every regionseparately.

4. Experimental Results for Range Estimation

In this section we use the axial variation of a pro-jected speckle pattern to estimate the axial locationof a large object. The experiment of range estimationwas performed in reflective configuration in whichthe ground glass (diffuser) was used to project thespeckle pattern on top of a reflective diffuse object,and the reflection was imaged on a CCD camera.

Fig. 4. (Color online) Captured image containing (a) the inputpattern that contains a structure of transmissive glasses withwidths of zero, one, and two slides (each slide is 0:5mm thick),and (b) the image captured by the CCD camera when projectedwith speckles pattern.

Fig. 3. Calibration reference readout for the transmissive micro-scopic configuration of 3D mapping using speckles random coding.(a) Speckle pattern without any glass in the optical path.(b) Speckle pattern when one glass slide is inserted into the opticalpath. (c) Speckle pattern when two glass slides are inserted intothe optical path. (d) Speckle pattern when three glass slides areinserted into the optical path. Each slide is 0:5mm thick. The in-sets show a magnified portion of the same region for each figure.

Fig. 5. Segmentation of different regions by absolute value ofsubtraction between the captured patterns and the reference pat-terns. (a) Segmentation of Region 0. (b) Segmentation of Region 1.(c) Segmentation of Region 2.


The system is shown in Fig. 6. Light from a lasersource passes through the diffuser. The speckle pat-terns coming from that diffuser are projected on thereflective object and recorded by a CCD camera. Theeffective size of the diffuser is adjusted by means ofthe diaphragm until the image of the projectedspeckle size is larger than the camera pixel for cor-rect recording. Thus for the case of reflected config-uration, the optical parameters are such that thespeckle statistics coming from the diffuser (creatingthe primary projected speckle pattern) are the domi-nant rather than the influence of the statistics of theobject texture itself (creating the imaging specklepattern as described in Section 2). Such a situationmay be obtained, for instance, by choosing the properdiameter of the spot that illuminates the diffuser(since this diameter determines the size of the pro-jected speckles) such that it generates primaryspeckles that are larger than the secondary specklesgenerated due to the transmission from the object.That way if the diameter of the imaging lens is largerthan the diameter of the effective diameter of thediffuser, the primary speckles are fully seen by thediffraction resolution limit of the camera (the Fnumber), while the secondary speckles, which aresmaller, will be partially filtered by the camera sinceits F number will be too large to contain their fullspatial structure. In our experiments we use a lenswith an F number of 4, which, for the 532 laser, givesan imaging speckle size of ∼2 μm on the CCD, wellbelow the camera resolution.For range finding the object was positioned 80 cm

away from the CCD camera. The diffuser diameter isset to get an illumination F number of approximately40, which corresponds to an axial decorrelationlength of 6:8mm according to Eq. (9). To capturethe decorrelated speckle patterns, a calibration pro-cess was done. It consists of mapping the object bycapturing images reflected from 11 planes separatedby 5mm. For quantifying the speckle pattern varia-tions, we use conventional correlation (integral of theproduct). Initially we record the reference images of

the speckle pattern of a diffuse object (a matte whitepainted plate) at a set of axial distances. We take the11 reference images, each one separated axially by5mm. This set of images is indeed a 3D map ofthe speckle pattern in the volume of interest. Thenwe move again on the object in the same axial dis-tance range and obtain the correlation of every imagewith all the reference set. Therefore we obtain a col-lection of 11 × 11 correlation values between the 11patterns at different distances and the 11 referenceimages, covering a total axial range of 55mm.

As an example, in Fig. 7(a) we present the reflectedreference speckle pattern that corresponds to a planepositioned 80 cm away from the camera. Note thediagonal black line, which we use as a referencefor focusing and positioning. In Fig. 7(b) we depictthe autocorrelation between the reflected specklepattern from a certain plane and its reference distri-bution obtained in the calibration process. Two maincomponents can be observed in the autocorrelation.On one hand we have the sharp, intense, small peakin the center of the image, which corresponds to thespeckle correlation. On the other hand the bulk of thepattern produces a long, broad diagonal correlationpattern owing to the dark line in the sample. Thesmall correlation peak is significantly higher thanthe broad one. In Fig. 7(c) we present the cross cor-relation of speckle patterns reflected from two planesseparated by 5mm. Note that the small correlationpeak, due to the fine details of the image, has inten-sity similar to the broad bulk correlation signal, qua-litatively showing the decrease of the autocorrelationdue to speckle pattern as the two correlated planesare separated axially.

In Fig. 8 we show the range finding results, show-ing the correlations of each image with the calibra-tion set. As one may see the maxima appear only

Fig. 7. Reflective speckle patterns used for range finding. (a) Thereflected reference speckle pattern corresponds to a plane posi-tioned 80 cm from the camera. The inset is a magnified region.(b) An autocorrelation between the reflected speckles pattern froma certain plane. (c) Cross correlation of speckle patterns reflectedfrom two planes separated by 5mm.

Fig. 6. (Color online) Optical setup for the range estimation.


at the diagonal (autocorrelation), while the distribu-tion outside the diagonal is small (cross correlation).For further clarification Fig. 8 is actually a cross cor-relation matrix of 11 × 11, while the indexes (1 to 11)on the horizontal axes represent the plane numberand the vertical axis represents the correlationvalue. The diagonal of the matrix is the correlationvalue of two images taken at the same distance.Although we have termed these cases as autocorre-lation, we want to point out that the correlation isbetween two very similar images but taken in differ-ent instants of time and after relocation of the axialand lateral position. The nondiagonal terms are re-lated to the cross correlation values between the 11planes. The neat intensity separation between theautocorrelation values and the cross correlationvalues permits us to identify the distance of the pat-tern by cross correlation of the pattern at a given un-known distance with the prerecorded patterns.Note that the process does not depend on the mi-

crostructure of the pattern because the ranging isperformed on the projected speckle pattern andnot on the imaging one. The spectrum of the capturedimage with only imaging speckle (with uniform illu-mination of the object) is basically white because theNyquist frequency of the CCD is much smaller thanthe typical frequency of the pattern. The effect of theimaging speckle is at pixel level and reduces the sig-nal-to-noise ratio in the projected pattern compari-son. Because of this fact the object itself is notrelevant, provided that it is a fine diffuse object.As a confirmation of this fact, the region of the testplate used for the experiments was not the same inthe calibration images as in the test image, thus hav-ing a different microstructure in the field of view.

5. Experimental Results of the Three-DimensionalMapping of Reflective Objects

In Section 4 we used the large area correlation for thedepth determination. In this section we propose theuse of local correlation to map the depths of small re-

gions in a 3D object. We use a calibration proceduresimilar to the previous one, capturing the specklepattern projected on a white painted plate at a setof axial positions that cover the working volume.In this case the correlation is not performed forthe full image as in the range finding case, but ratherwe perform a correlation of the local windows of theobject with the same region of each of the calibrationpatterns to give the final 3D mapping. Note that inthis case the projected speckle pattern is capturefrom the surface of completely different objects dur-ing the calibration (reference plane diffuse plate) andduring the mapping (the object itself).

A basic parameter that we need for the procedureis the minimum size of the correlation window thatwill separate the autocorrelation value of the specklepattern from the cross correlation between two win-dows with uncorrelated (by depth changes) speckles.A analytical treatment is possible (see Chap. 4 inRef. [12] for a similar problem), but the solution isgenerally not in a closed form and may be influencedby additional factors such as the digitalization noise.Therefore we prefer to perform a preliminary test tocheck the minimum window size. We use one of thereference images obtained by projecting the specklepattern on a plane plate. We perform autocorrela-tions and cross correlations between windows chosenat random positions in the field. The mean and thevariance of the correlations are obtained for 500 po-sitions. The results are plotted in Fig. 9, showing themean values and the confidence interval for one stan-dard deviation for the autocorrelation for differentcircular window sizes normalized to the speckle size.As can be seen in the figure, the two values can beseparated from window sizes exceeding 3.5 timesthe speckle size. In the following experiment we useda running window size of 10 pixels.

In Fig. 10 we show the results that were obtainedfor a 3D object made up with toy building blocks. The

Fig. 8. Range finding results. The cross correlation intensities ofthe combinations of the reflected patterns from all possible 11planes positioned at a regular spacing of 5mm.

Fig. 9. Variation of the mean and confidence interval at one stan-dard deviation for the autocorrelation and cross correlation ofvarying size of circular regions.


object occupied a volume of 40 × 25 × 30mm. As amain advantage over most 3D mapping methods,the proposed procedure has virtually no shadowingbecause the illumination and image recording areperformed from the same location.

6. Conclusions

We have presented the optical usage of speckles forthe thickness mapping of transparent objects, 3D dif-fuse object mapping, and range estimation. The keyconcept is to use a ground glass that produces apattern that changes with the depth inherently byfree-space propagation. The projected pattern dec-orrelates when it is sampled at two different posi-tions separated either axially or transversely. The

computation of the correlation between the actualpattern as projected or transmitted by the objectand a set of calibration images allow the determina-tion of the 3D location or shape of the object. Theconcept has also been applied to 3D mapping by com-puting the correlation between the object and the re-ference images on a running window of small size.Despite the relatively small resolution, the systemis robust and almost independent on the character-istics of the object. The projection of the speckle inthe volume can be made from a small and thus solidand stable system because it comprises only the laserand the diffuser. A significant advantage of themethod is the absence of shadowed area and highconfigurability. Experimental results present theverification of the proposed directions.

This work was partially supported by the FEDERfunds Spanish Ministerio de Educación y Cienciaunder the project FIS2007-60626.

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Fig. 10. (a) Object illuminated with the projected specklespattern. (b) 3D reconstruction.


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three-dimensional mapping and range measurement by means of projected speckle patterns

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